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1

FLOW SYSTEMS

1.1 CONSTRUCTAL LAW, VASCULARIZATION, AND SVELTENESS

A flow represents the movement of one entity relative to another (the background).To describe a flow, we speak of what the flow carries (fluid, heat, or mass), howmuch it carries (mass flow rate, heat current, etc.), and where the stream is located.The where is the focus of this new course. A flow system has configuration, drawing,that is, design.

In science, the origin (genesis) of the configuration of flow systems has beenoverlooked. Design has been taken for granted—at best, it has been attributed tochance, inspiration, talent, and art. Our own education in the sciences is based onsketches of streams into and out of boxes, sketches that bear no relation to reality,to the position that the stream occupies in space and in time. This book is ourattempt to change this attitude.

The benefits from thinking of design as science are great. The march towardsmaller dimensions (micro, nano)—the miniaturization revolution—is not aboutmaking smaller and smaller components that are to be dumped like sand into asack. This revolution is about the “living sack,” in which every single component iskept alive with flows that connect it to all the other components. Each componentis put in the right place, like the neurons in the brain, or the alveoli in the lung. Itis the configuration of these extremely numerous components that makes the sackperform at impressively high levels.

Because natural flow systems have configuration, in this book we treat theemergence of flow configuration as a physics phenomenon that is based on a scien-tific principle. Constructal theory is the mental viewing that the generation of theflow structures that we see everywhere in nature (river basins, lungs, atmospheric

1

COPYRIG

HTED M

ATERIAL



2 Flow Systems

circulation, vascularized tissues, etc.) can be reasoned based on an evolutionaryprinciple of increase of flow access in time, i.e. the time arrow of the animatedmovie of successive configurations. That principle is the constructal law [1–4]:

For a finite-size flow system to persist in time (to live), its configuration must changein time such that it provides easier and easier access to its currents (fluid, energy,species, etc.).

Geometry or drawing is not a figure that always existed and now is availableto look at, or worse, to look through and take for granted. The figure is the per-sistent movement, struggle, contortion, and mechanism by which the flow systemachieves global objective under global constraints. When the flow stops, the figurebecomes the flow fossil (e.g., dry river bed, snowflake, animal skeleton, abandonedtechnology, and pyramids of Egypt).

What is the flow system, and what flows through it? These are the questions thatmust be formulated and answered at the start of every search for architectures thatprovide progressively greater access to their currents. In this book, we illustratethis thinking as a design method, mainly with examples from engineering. Themethod, however, is universally applicable and has been used in a predictive senseto predict and explain many features of design in nature [1–4, 10].

Constructal theory has brought many researchers and educators together, on sev-eral campuses (Duke; Toulouse; Lausanne; Evora, Portugal; Istanbul; St. John’s,Newfoundland; Pretoria; Shanghai) and in a new direction: to use the constructallaw for better engineering and for better organization of the movement and con-necting of people, goods, and information [2–4]. We call this direction constructaldesign, and with it we seek not only better configurations but also better (faster,cheaper, direct, reliable) strategies for generating the geometry that is missing.

For example, the best configurations that connect one component with verymany components are tree-shaped, and for this reason dendritic flow architecturesoccupy a central position in this book. Trees are flows that make connectionsbetween points and continua, that is, infinities of points, namely, between a volumeand one point, an area and one point, and a curve and one point. The flow mayproceed in either direction, for example, volume-to-point and point-to-volume.

Trees are not the only class of multiscale designs to be discovered and used. Wealso teach how to develop multiscale spacings that are distributed nonuniformlythrough a flow package, flow structures with more than one objective, and, espe-cially, structures that must perform both flow and mechanical support functions.Along this route, we unveil designs that have more and more in common withanimal design. We do all this by invoking a single principle (the constructal law),not by copying from nature.

With “animal design” as an icon of ideality in nature, the better name for theminiaturization trends that we see emerging is vascularization. Every multiscalesolid structure that is to be cooled, heated, or serviced by our fluid streams must beand will be vascularized. This means trees and spacings and solid walls, with everygeometric detail sized and positioned in the right place in the available space. These



1.1 Constructal Law, Vascularization, and Svelteness 3

will be solid-fluid structures with multiple scales that are distributed nonuniformlythrough the volume—so nonuniformly that the “design” may be mistaken asrandom (chance) by those who do not quite grasp the generating principle, justlike in the prevailing view of animal design, where diversity is mistaken forrandomness, when in fact it is the fingerprint of the constructal law [1–4].

We see two reasons why the future of engineering belongs to the vascularized.The first is geometric. Our “hands” (streams, inlets, outlets) are few, but they mustreach the infinity of points of the volume of material that serves us (the devices,the artifacts, i.e. the engineered extensions of the human body). Point-volume andpoint-area flows call for the use of tree-shaped configurations. The second reasonis that the time to do such work is now. To design highly complex architectures oneneeds strategy (theory) and computational power, which now we possess.

The comparison with the vascularization of animal tissue (or urban design,at larger scales) is another way to see that the design philosophy of this bookis the philosophy of the future. Our machines are moving toward animal-designconfigurations: distributed power generation on the landscape and on vehicles,distributed drives, distributed refrigeration, distributed computing, and so on. Allthese distributed schemes mean trees mating with trees, that is, vascularization.

A flow system (or “nonequilibrium system” in thermodynamics; see Chapter 2)has new properties that are complementary to those recognized in thermodynamicsuntil now. A flow system has configuration (layout, drawing, architecture), whichis characterized by external size (e.g., external length scale L), and internal size(e.g., total volume of ducts V , or internal length scale V1/3). This means that a flowsystem has svelteness, Sv, which is the global geometric property defined as [5]

Sv = external flow length scale

internal flow length scale(1.1)

This novel concept is important because it is a property of the global flow architec-ture, not flow kinematics and dynamics. In duct flow, this property describes therelative importance of friction pressure losses distributed along the ducts and localpressure losses concentrated at junctions, bends, contractions, and expansions. Itdescribes the “thinness” of all the lines of the drawing of the flow architecture (cf.Fig. 1.1).

To illustrate the use of the concept of svelteness, consider the flow through twoco-linear pipes with different diameters, D1 < D2 (Fig. 1.2). The pipe lengths are L1

and L2. The sudden enlargement of the flow cross-section leads to recirculation anddissipation (imperfection, Chapter 2) immediately downstream of the expansion.This effect is measured as a local pressure drop, which in Example 1.1 (p. 16) isderived by invoking the momentum theorem:

�Plocal =[

1 −(

D1

D2

)2]2

1

2ρV 2

1 (1.2)



4 Flow Systems

Svelteness (Sv) increases

Figure 1.1 The svelteness property Sv of a complex flow architecture: Sv increases from leftto right, the line thicknesses decrease, and the drawing becomes sharper and lighter, that is moresvelte. The drawing does not change, but its “weight” changes.

Equation (1.2) is known as the Borda formula. In the calculation of total pressurelosses in a complex flow network, it is often convenient to neglect the local pressurelosses. But is it correct to neglect the local losses?

The calculation of the svelteness of the network helps answer this question. Thesvelteness of the flow geometry of Fig. 1.2 is

Sv = L1 + L2

V 1/3(1.3)

0 10 20 30 40 500.0

0.5

1

Sv

local

distributed

�P�P

Turbulent fully developed (f � 0.01)Re � 102 103

Laminar fully developed

Figure 1.2 The effect of svelteness (Sv) on the importance of local pressure losses relative todistributed friction losses in a pipe with sudden enlargement in cross-section (cf. Example 1.1).



1.1 Constructal Law, Vascularization, and Svelteness 5

Surface condition ks [mm]

Riverted stell Concrete Wood staveCast ironGalvanized iron

0.9-90.3-30.18-0.90.260.15

Asphalted cast ironCommercial steel or wrought ironDrawn tubing

0.120.050.0015

0.05

0.1

0.05

0.04

0.03

0.02

0.01

103 104

Smooth pipes(the Karman-Nikuradse relation)

Laminar flow,

105 106 107 108

0.040.03

0.020.0150.010.0080.0060.004

0.002

0.0010.00080.00060.00040.00020.00010.000,05

0.000,001

0.000,0050.000,001

Rel

ativ

e ro

ughn

ess k s

/D

ReD � UD/v

4f

16ReD

f�

Figure 1.3 The Moody chart for friction factor for duct flow [6].

where V is the total flow volume, namely V = (π/4)(D21 L1 + D2

2 L2). The dis-tributed friction losses (�Pdistributed) are associated with fully developed (laminaror turbulent) flow along L1 and L2, and have the form given later in Eq. (1.22), forwhich the friction factors are furnished by the Moody chart (Fig. 1.3).

The derivation of the curves plotted in Fig. 1.2 is detailed in Example 1.1. Theratio �Plocal/�Pdistributed decreases sharply as Sv increases. When Sv exceeds theorder of 10, local losses become negligible in comparison with distributed losses.We retain from this simple example that Sv is a global property of the flow space“inventory.” This property guides the engineer in the evaluation of the performanceof the flow design.

A flow system is also characterized by performance (function, objective, direc-tion of morphing). Unlike in the black boxes of classical thermodynamics, a flowsystem has a drawing. The drawing is not only the configuration (the collection ofblack lines, all in their right places on the white background), but also the thick-nesses of the black lines. The latter gives each line its slenderness, and, in ensemble,the slendernesses of all the lines account for the svelteness of the drawing.



6 Flow Systems

The constructal design method guides the designer (in time) toward flow archi-tectures that have greater and greater global performance for the specified flowaccess conditions (fluid flow, heat flow, flow of stresses). The architecture discov-ered in this manner for the set of conditions “1” is the “constructal configuration1”. For another set of conditions, called “2”, the method guides the designer to the“constructal configuration 2”. In other words, a configuration developed for one setof conditions is not necessarily the recommended configuration for another set ofconditions. The constructal configuration “1” is not universal—it is not the solutionto other design problems. Universal is the constructal law, not one of its designs.

For example, we will learn in Fig. 4.1 that the best way to size the diametersof the tubes that make a Y-shaped junction with laminar flow is such that the ratioof the mother/daughter tube diameters is 21/3. The geometric result is good formany flow architectures that resemble Fig. 4.1, but it is not good for all situationsin which channels with two diameters are present. For example, the 21/3 ratio isnot necessarily the best ratio for the large/small diameters of the parallel channelsillustrated in Problem 3.7. For the latter, a different search for the constructalconfiguration must be performed, and the right diameter ratio will emerge at theend of that search.

In this chapter and the next, we review the milestones of heat and fluid flowsciences. We accomplish this with a run through the main concepts and results,such as the Poiseuille and Fourier formulas. We do not derive these formulas fromthe first principles, for example, the Navier-Stokes equations (F = ma) and firstlaw of thermodynamics (the energy conservation equation). Their derivation is theobject of the disciplines on which the all-encompassing and new science of designrests.

In this course we do even better, because in the analysis of various configurationswe derive the formulas for how heat, fluid, and mass should flow. In other words,we teach the disciplines with purpose, on a case-by-case basis, not in the abstract.We teach the disciplines for a second time.

In this review the emphasis is placed on the relationships between flow ratesand the “forces” that drive the streams. These are the relations that speak ofthe flow resistances that the streams overcome as they flow. In Chapter 2 wereview the principles of thermodynamics in order to explain why streams thatflow against resistances represent imperfections in the greater flow architecturesto which they belong. The body of this course and book teaches how to distributethese imperfections so that the global system is the least imperfect that it can be.The constructal design method is about the optimal distribution of imperfection.

1.2 FLUID FLOW

In this brief review of fluid mechanics we assume that the flow is steady and thefluid is newtonian with constant properties. Each flow example is simple enoughso that its derivation may be pursued as an exercise, in the classroom and as



1.2 Fluid Flow 7

homework. Indeed, some of the problems proposed at the end of chapters are ofthis type. Our presentation, however, focuses on the formulas to use in design, andon the commonality of these flows, which in the case of duct flow is the relationbetween pressure difference and flow rate. The presentation proceeds from thesimple toward more complex flow configurations.

1.2.1 Internal Flow: Distributed Friction Losses

We start with the Bernoulli equation, written for perfect (frictionless) fluid flowalong a streamline,

P + ρgz + 1

2ρV 2 = constant (1.4)

where P, z, and V are the local pressure, elevation, and speed. For real fluid flowfrom cross-section 1 to cross-section 2 of a stream tube, we write

P1 + ρgz1 + 1

2ρV 2

1 = P2 + ρgz2 + 1

2ρV 2

2 + �P (1.5)

where �P represents the sum of the pressure losses (the fluid flow imperfection)that occurs between cross-sections 1 and 2. The losses �P may be due to distributedfriction losses, local losses (junctions, bends, sudden changes in cross-section), orcombinations of distributed and local losses (cf. Example 1.1).

Fully developed laminar flow occurs inside a straight duct when the duct issufficiently slender and the Reynolds number sufficiently small. For example, ifthe duct is a round tube of inner diameter D, the velocity profile in the ductcross-section is parabolic:

u = 2U

[1 −

(r

r0

)2]

(1.6)

In this expression, u is the longitudinal fluid velocity, U is the mean fluid velocity(i.e., u averaged over the tube cross-section), and r0 is the tube radius, r0 = D/2. Inhydraulic engineering, more common is the use of the volumetric flowrate Q[m3/s],which for a round pipe is defined as

Q = Uπr20 (1.7)

The radial position r is measured from the centerline (r = 0) to the tube wall(r = r0). This flow regime is known as Hagen-Poiseuille flow, or Poiseuille flowfor short. The derivation of Eq. (1.6) can be found in Ref. [6], for example.

How the pressure along the tube drives the flow is determined from a globalbalance of forces in the longitudinal direction. If the tube length is L and thepressure difference between entrance and exit is �P, then the longitudinal force



8 Flow Systems

balance is

�Pπr20 = τ2πr0L (1.8)

The fluid shear stress at the wall is

τ = μ

(−∂u

∂r

)r=r0

(1.9)

Equations (1.8) and (1.9) are valid for laminar and turbulent flow. By combiningEqs. (1.6) through (1.9), we conclude that for laminar flow the mean velocity isproportional to the longitudinal pressure gradient

U = r20

8μ

�P

L(1.10)

Alternatively, we use the mass flow rate

m = ρUπr20 (1.11)

to rewrite Eq. (1.10) as a proportionality between the “across” variable (�P) andthe “through” variable (m):

�P = mL

r40

8

πν (1.12)

The tandem of “across” and “through” variables has analogues in other flow sys-tems, for example, voltage and electric current, temperature difference and heatcurrent, and concentration difference and flow rate of chemical species (see Chap-ter 2, Fig. 2.4). In Eq. (1.12), the ratio �P/m is the flow resistance of the tubelength L in the Poiseuille regime. This resistance is proportional to the geometricgroup L/r4

0 , or L/D4:

�P

m= L

D4

128

πν (1.13)

The flow is laminar provided that the Reynolds number

ReD = U D

ν(1.14)

is less than approximately 2000. The tube length L is occupied mainly by Poiseuilleflow if L is greater (in order of magnitude sense) than the laminar entrance lengthof the flow, which is X � DReD [6]. The condition for a negligible entrance lengthis therefore

L

D� ReD (1.15)



1.2 Fluid Flow 9

The flow resistance solution (1.12) has been recorded alternatively in terms of afriction factor, which is defined as

f = τ12ρU 2

(1.16)

After using Eqs. (1.6) and (1.9), the friction factor for Poiseuille flow through around tube becomes

f = 16

ReD(1.17)

The Poiseuille flow results for straight ducts that have cross-sections other thanround are recorded in a notation that parallels what we have just reviewed forround tubes. Let A and p be the area and wetted perimeter of the arbitrary ductcross-section. The average fluid velocity U is defined as

ρ AU =∫∫

ρu d A = ρQ (1.18)

Instead of Eq. (1.8), we have the force balance

�P · A = τpL (1.19)

Instead of ReD, we use the Reynolds number based on the hydraulic diameter ofthe general cross-section,

ReDh = U Dh

ν(1.20)

Dh = 4A

p(1.21)

Laminar fully developed flow prevails if ReDh ≤ 2000. The duct length L is sweptalmost entirely by Poiseuille flow if L/Dh � ReDh , cf. Eq. (1.15).

The friction factor definition (1.16) continues to hold. Now, if we combine Eqs.(1.16) and (1.19) through (1.21), we arrive at the general pressure drop formula

�P = f4L

Dh

1

2ρU 2 (1.22)

For Poiseuille flow, the friction factor assumes the general form

f = Po

ReDh

(1.23)

where Po is the Poiseuille constant, for example Po = 16 for round tubes, andPo = 24 for channels formed between parallel plates. We see that every cross-sectional shape has its own Poiseuille constant, and that this constant is not muchdifferent than 16. A channel with square cross-section has Po = 14.2, and onewith equilateral triangular cross-section has Po = 13.3. We return to this subject inTable 1.2.



10 Flow Systems

A word of caution about the Dh and f definitions is in order. The factor 4 isused in Eq. (1.21) so that in the case of a round pipe of diameter D the hydraulicdiameter Dh is the same as D [substitute A = (π /4)D2 and p = πD in Eq. (1.21), andobtain Dh = D]. A segment of the older literature defines Dh without the factor 4,

D′h = A

p(1.24)

and this convention leads to a different version of Eq. (1.22):

�P = f ′ L

D′h

1

2ρU 2 (1.25)

The alternate friction factor f ′ obeys the definition (1.16). We also note thatDh = 4 D′

h and f ′ = 4f , such that for a round pipe with diameter D and Poiseuilleflow the formulas are f = 16/ReD and f ′ = 64/ReD. The numerators (16 vs. 64)are the first clues to remind the user which Dh definition was used, Eq. (1.21) orEq. (1.24). The 4f plotted on the ordinate of Fig. 1.3 suggests that this chart wasoriginally drawn with f ′ on the ordinate.

To summarize, by combining Eqs. (1.22) and (1.23) we conclude that all thelaminar fully developed (Poiseuille) flows are characterized by a proportionalitybetween �P and U, or �P and m:

�P

m= 2Poν

L

D2h A

(1.26)

Verify that for a round tube (Po = 16, Dh = D, A = πD2/4), Eq. (1.26) leadsback to Eq. (1.23). The general proportionality (1.26) is a straight line drawn atReDh < 2000 in Fig. 1.3 – one line for each tube cross-sectional shape. In Fig. 1.3,only the line for the round tube is shown. The Moody chart is highly useful becauseit is a compilation of many empirical formulas (f vs. ReDh) in a single place–abird’s-eye view of distributed friction losses in all ducts with fully developedlaminar or turbulent flow.

For turbulent flow through the same ducts, the �P vs. m relation is different. Onerelation that is independent of the flow regime is the pressure drop formula (1.22),because this formula is a rewriting of the force balance (1.19) in combination withthe friction factor definition (1.16). Different for turbulent flow is the manner inwhich f depends on the flow rate, or ReDh .

The Moody chart (Fig. 1.3) shows that in turbulent flow f is a function ofReDh and the roughness of the wall inner surface. Plotted on the abscissa is ReD

(round tube), however, the curves for turbulent flow hold for all cross-sectionalshapes provided that ReD is replaced by ReDh . To assist computer-aided design andanalysis, the friction factor function f (ReD, ks/D) is available in several alternative



1.2 Fluid Flow 11

mathematical forms. The Colebook relation is often used for turbulent flow:

1

f 1/2= −4 log

(ks/D

3.7+ 1.256

f 1/2ReD

)(1.27)

Because this expression is implicit in f , iteration is required to obtain the frictionfactor for a specified ReD and ks/D. Several explicit approximations are availablefor smooth ducts:

f ∼= 0.079Re−1/4D (2 × 103 < ReD < 105) (1.28)

f ∼= 0.046Re−1/5D (2 × 104 < ReD < 106) (1.29)

At sufficiently high Reynolds numbers, the friction factor depends only on therelative roughness, which means that for a given duct f is constant. This is knownas the fully rough regime. This feature and Eq. (1.22) show that in fully roughturbulent flow �P is proportional to U2, or to m2,

�P

m2= 2 f

ρ

L

Dh A2(1.30)

Compare this fully rough turbulent resistance with the fully developed laminarresistance (1.26), and you will see in another way the fact that the slope of thecurves changes from left to right in Fig. 1.3.

1.2.2 Internal Flow: Local Losses

The relations between pressure drops and flow rates become more complicatedwhen the ducts are too short to satisfy the negligible entrance assumption (1.15),and when they are connected with other ducts into larger flow networks (e.g., Fig.1.4). The treatment of such cases is based on a formulation that begins with thegeneralized Bernoulli equation for irreversible flow through a duct as a controlvolume, from entrance (1) to exit (2), as shown in Eq. (1.5). The pressure loss �Pis calculated by summing up all the flow imperfections,

�P =∑

d

[f

4L

Dh

1

2ρU 2

]d︸︷︷︸

distributed losses

+∑

l

[K

1

2ρU 2

]l︸︷︷︸

local losses

(1.31)

The first summation refers to sections of long ducts, along which the pressure dropsare due to fully developed flow, laminar or turbulent. Note the similarity betweenthe expression written inside [ ]d and Eq. (1.22).

The second summation in Eq. (1.31) refers to local losses such as the pressuredrops caused by junctions, fittings, valves, inlets, outlets, enlargements and con-tractions. Each local loss contributes to the total loss in proportion to KρU2/2,where K is the respective local-loss coefficient. Experimental K data are available



12 Flow Systems

L3

L4

L2

L1Pump

90� elbow

(a)

(b)

a b

1

2

3

Figure 1.4 Examples of networks of pipes: (a) single flow path; (b) multiple paths [7].

in the fluids engineering literature: Table 1.1 shows a representative sample. Locallosses can be non-negligible in the functioning of complex tree-shaped networks.How important, and when they can be neglected are questions tied closely to theconcept of flow architecture svelteness, as we show in Example 1.1. It suffices tosay that as the svelteness of the flow architecture increases from left to right inFig. 1.1, all the lines of the drawing become thinner and, consequently, the dis-tributed losses gain in importance relative to the local (junctions) losses.

Most of the fluid flow examples in this book have the objective to minimize intime the “energetic consumption” (exergy destruction) of the fluid network. Wewill now see that this objective can be attained by minimizing the overall pressurelosses �P. We show this for an open system, and later for a closed system.

Assume that the open flow system is an urban hydraulic network. A certainvolume of water V must be delivered per day to a city. This water is pumped froma river or a lake. Some water treatment may be needed to make the water drinkablefor the users: this water treatment will occur at the level of the pump stations. Thepumped water is not sent directly to the users in the city. The preferred solutionconsists of placing a water tank at a height between the pumping station and thecity. In such a system, the tank has a double role. The first is to store water sothat in case of pump malfunction the water supply (aqueduct) is decoupled from



1.2 Fluid Flow 13

Table 1.1 Local loss coefficients [7].

Resistance K

Changes in Cross-Sectional Areaa

Round pipe entrance

0.04–0.28

Contraction

AR = smaller arealarger area 0.45(1−AR)

θ < 60◦

0.1 ≤ AR ≤ 0.50.04–0.08

Expansion

(1 − 1

AR

)2

1.0

Valves and FittingsGate valve, openGlobe valve, openCheck valve (ball), open45-degree elbow, standard90-degree elbow, rounded

0.26–10

700.3–0.40.4–0.9

a The velocity used to evaluate the loss is the velocity upstream of the expansion. WhenAR = 0, K = 1.

the distribution network, and the city is not affected by accidents and continues toreceive drinkable water. The second role of the tank is to provide a controlled waterpressure to the city simply because of the altitude difference.

Figure 1.5 shows the main features of a water supply network. The total volumeof water pumped per day and stored in the tank is fixed. The objective is to minimizethe energetic consumption of the network C. The role of the pumping station (Hm)in this network is to compensate for the altitude difference between the tank and



14 Flow Systems

Source

Pump

Tank

V

zg

0

Figure 1.5 The route of city water supply from the source to the storage tank.

the source (river, lake) and the overall pressure loss,

Hm = ztank − zsource + ξ (1.32)

Here, ξ is the overall pressure loss expressed as head of water, that is, the height ofa column of water, ξ = �P/ρg, where �P is the overall pressure loss. Typically,the pumps used in urban hydraulic applications are centrifugal pumps with bladescurved backward. Their characteristic is given in Fig. 1.6 together with the networkhead curve [see the right-hand side of (1.32)]. The intersection of the two curvesis called the operating point. The energetic consumption of the system is given bythe product of the pumping power and the time of pumping,

C = W t (1.33)

where

W = mgHm (1.34)

C = ρgHm V (1.35)

H

m0

0

Net head change: head produced by the pump (Hm)

Operating point

Characteristic curve of thewater distribution network (ztank� zsource � �)

Figure 1.6 The operating point of the water distribution network, as the intersection betweenthe characteristic curve of the pump and the characteristic curve of the water distribution network.



1.2 Fluid Flow 15

H

m0

0

Pressure losses decrease

Figure 1.7 The change in the operatingpoint as the pressure losses of the networkdecrease.

Because Hm ∼ �P and the water volume V is constrained, we see that C ∼ �P.Therefore, to minimize the energetic consumption means to minimize the pressurelosses. Figure 1.7 shows how the operating point shifts when the pressure lossesare minimized. A higher flow rate is the result.

Next, we connect the tank to the users (Fig. 1.8). No pump is needed because thepressure required at the city level is maintained by the altitude difference betweenthe tank and the city,

ρgztank = Prequired + ρgzcity + �P (1.36)

Note that the kinetic energy term 12ρU 2 does not appear here. The reason is that

in urban hydraulic applications the fluid velocity is in the range of 1 m/s, makingthe changes in kinetic energy negligible. Again, we see that the optimization ofthis flow system (a higher Prequired) means that the pressure losses �P must beminimized.

An example of a closed system is presented in Fig. 1.9, which shows the principleof steady-flow operation of a heating network in a building. The network is madeof basic components such as a water heater, a radiator, and a pump. The heateruses combustion in order to heat the water stream. The radiator releases heat intothe space of the building. Because the system is closed, the gravity effect vanishes(the water that flows up must eventually flow down). Thus, we arrive at Hm ∼ �P.

Source

Pump

Tank

V

City, users

Figure 1.8 Complete network for water supply, storage, and distribution.



16 Flow Systems

Radiator

Waterheater

Pump

Figure 1.9 Closed loop circulation of water forheating a building.

Once again, the pumping power will be minimized when the pressure losses areminimized.

Example 1.1 Here, we show how to determine the local pressure loss associ-ated with the sudden expansion of a duct (Fig. 1.2). Incompressible fluid flowsthrough a pipe of diameter D1 and length L1, and continues flowing througha pipe of diameter D2 and length L2. Consider the cylindrical control volumeindicated with dashed line in Fig. 1.10. The control-surface convention that weuse is that there is no space between the dashed line and the wall (the solidline). The dashed line represents the internal surface of the wall. The controlvolume contains the enlargement of the stream, including the recirculation oc-curring after the backward-facing steps. We apply the momentum theorem inthe longitudinal direction,

m(V1 − V2) = A2(P2 − P1) (a)

where m is the mass flow rate

m = ρ A1V1 = ρ A2V2 (b)

and A1,2 = (π/4)D21,2. See the lower part of Fig. 1.10. From Eq. (a) we learn

that the pressure at the outlet of the control volume is

P2 = P1 + ρV2(V1 − V2) (c)

This pressure can be compared with the value in the ideal (frictionless, re-versible) limit where the stream expands smoothly from A1 to A2. According tothe Bernoulli equation,

P1 + ρgz + 1

2ρV 2

1 = P2 + ρgz + 1

2ρV 2

2 + �Plocal (d)



1.2 Fluid Flow 17

P1

1 V2

2P

2 2L , D

2A

1 1P A 2 2P A

1mV 2mVControl volumeImpulse Reaction

V1

L1, D1

A1

Figure 1.10 Control volume formulation for the duct with sudden expansion and localpressure loss, which is analyzed in Example 1.1.

the local pressure loss is

�Plocal = P1 − P2 + 1

2ρ(V 2

1 − V 22 ) (e)

which, after using Eq. (c) yields

�Plocal = 1

2ρ(V1 − V2)2 (f)

Next, we eliminate V2 using Eq. (b), and arrive at the local loss coefficientreported in Table 1.1 [see also Eq. (1.2)]:

K = �Plocal12ρV 2

1

=(

1 − A1

A2

)2

=[

1 −(

D1

D2

)2]2

(g)

To show analytically how the local loss becomes negligible as the sveltenessof the flow system increases, assume that the small pipe is much longer than thewider pipe. The Sv definition (1.1) becomes

Sv ∼= L1(π

4D2

1 L1

)1/3 =(

4

π

)1/3 (L1

D1

)2/3

(h)



18 Flow Systems

The distributed losses are due to fully developed flow in the L1 pipe. Accordingto Eq. (1.22), the pressure drop along L1 is

�P1 = f14L1

D1

1

2ρV 2

1 = �Pdistributed (i)

Dividing Eqs. (g) and (i) we obtain

�Plocal

�Pdistributed= 1

4 f

[1 − (D1/D2)2

]2

L1/D1(j)

and, after using Eq. (h),

�Plocal

�Pdistributed= 1

4 f

[1 − (D1/D2)2

]2

(π/4)1/2Sv3/2 (k)

The ratio �Plocal/�Pdistributed decreases as Sv increases. If the flow in theL1 pipe is in the fully turbulent and fully rough regime, then f is a constant(independent of Re1) with a value of order 0.01. For simplicity, we assumef ∼= 0.01 and (D1/D2)2 � 1, such that Eq. (k) yields the curve plotted forturbulent flow in Fig. 1.2:

�Plocal

�Pdistributed

∼= 25

Sv3/2 (l)

Noteworthy is the criterion for negligible local losses, �Plocal � �Pdistributed,which according to Eq. (l) means Sv � 8.5.

If the flow in the L1 pipe is laminar and fully developed, then in Eq. (l) wesubstitute f = 16/Re1. The Reynolds number (Re1 = V1D1/ν) is an additionalparameter, which is known when m is specified. In place of Eq. (l) we obtain

�Plocal

�Pdistributed

Re1/64

Sv3/2 (m)

The criterion �Plocal � �Pdistributed yields in this case Sv � (Re1/64)2/3, whichmeans Sv � 6.2 when Re1 is of order 103. The curves for Re1 = 102 and 103

are shown in Fig. 1.2.Summing up, when the svelteness Sv exceeds 10 in an order of magnitude

sense, the local losses are negligible regardless of flow regime.

1.2.3 External Flow

The fins of heat exchanger surfaces, the trunks of trees, and the bodies of birds aresolid objects bathed all around by flows. Flow imperfection in such configurationsis described in terms of the drag force (FD) experienced by the body immersed in



1.2 Fluid Flow 19

100

10

10

1

1

1

0.1 0.1

CD

CD

10�1 102 103 104 105 1060.08

ReD

�v

U D �

Sphere

Sphere

Cylinderin cross-flow Cylinder

in cross-flow

Drag coefficient:

�FD /A

12

𝜌U2�

�

A � frontal area

FD � drag force

U � free-steramvelocity

Figure 1.11 Drag coefficients for smooth sphere and smooth cylinder in cross-flow [6].

a flow with free-stream velocity U∞,

FD = CD A f1

2ρU 2

∞ (1.37)

Here, Af is the frontal area of the body, that is, the area of the projection of thebody on a plane perpendicular to the free-stream velocity. The imperfection is alsomeasured as the mechanical power used in order to drag the body through the fluidwith the relative speed U∞, namely W = FDU∞. We return to this thermodynamicsaspect in Chapter 2.

The drag coefficient CD is generally a function of the body shape and theReynolds number. Figure 1.11 shows the two most common CD examples, for thesphere and the cylinder in cross-flow. The Reynolds number plotted on the abscissais based on the sphere (or cylinder) diameter D, namely ReD = U∞D/ν. These twobodies are the most important because with just one length scale (the diameter D)they represent the two extremes of all possible body shapes, from the most round(the sphere) to the most slender (the long cylinder).

Note that in the ReD range 102 – 105 the drag coefficient is a constant oforder 1. In this range FD is proportional to U 2

∞. In the opposite extreme, ReD < 10,the drag coefficient is such that the product CDReD is a constant. This small-ReD

limit is known as Stokes flow: here, FD is proportional to U∞.Compare side by side the flow resistance information for internal flow (Fig. 1.3)

with the corresponding information for external flow (Fig. 1.11). All the curves onthese log-log plots start descending (with slopes of −1), and at sufficiently highReynolds numbers in fully rough turbulent flow they flatten out. Said another way,



20 Flow Systems

f and CD are equivalent nondimensional representations of flow resistance, f forinternal flows, and CD for external flows.

1.3 HEAT TRANSFER

Heat flows from high temperature to lower temperature in the same way that a fluidflows through a pipe from high pressure to lower pressure. This is the natural way.The direction of flow from high to low is the one-way direction of the second law(Chapter 2). For heat flow as a mode of energy interaction between neighboringentities, the temperature difference is the defining characteristic of the interaction:heating is the energy transfer driven by a temperature difference.

Heat flow phenomena are in general complicated, and the entire discipline of heattransfer is devoted to determining the q function that rules a physical configurationmade of entities A and B [8]:

q = function (TA, TB, time, thermophysical properties, geometry, fluid flow)

(1.38)

In this section we review several key examples of the relationship between temper-ature difference (TA – TB) and heat current (q), with particular emphasis on the ratio(TA – TB)/q, which is the thermal resistance. Examples are selected because theyreappear in several applications and problems in this book. The parallels betweenthis review and the treatment of fluid-flow resistance (section 1.2) are worth noting.One similarity is the focus on the simplest configurations, namely, steady flow, andmaterials with constant properties.

1.3.1 Conduction

Conduction, or thermal diffusion, occurs when the two bodies touch, and there isno bulk motion in either body. The simplest example is a two-dimensional slabof surface A and thickness L. One side of the slab is at temperature T1 and theother at T2. The slab is the space and material in which the two entities (T1, T2)make thermal contact. The thermal conductivity of the slab material is k. The totalheat current across the slab, from T1 to T2, is described by the Fourier law of heatconduction,

q = kA

L(T1 − T2) (1.39)

The group kA/L is the thermal conductance of the configuration (the slab). Theinverse of this group is the thermal resistance of the slab,

Rt = T1 − T2

q= L

k A(1.40)



1.3 Heat Transfer 21

There is a huge diversity of body-body thermal contact configurations, andthermal resistances are available in the literature (e.g. Ref. [8]). As in the discussionof the sphere and the cylinder of Fig. 1.11, we cut through the complications ofdiversity by focusing on the extreme shapes, the most round and the most slender.One such extreme is the spherical shell of inner radius ri and outer radius ro. Wemay view this shell as the wrapping of insulation of thickness (ro – ri) on a sphericalhot body of temperature Ti, such that the outer surface of the insulation is cooledby the ambient to the temperature To. The thermal resistance of the shell is

Rt = Ti − To

q= 1

4πk

(1

ri− 1

ro

)(1.41)

where q is the total heat current from ri to ro, and k is the thermal conductivity ofthe shell material. Note that Eq. (1.41) reproduces Eq. (1.40) in the limit ro → ri,where the shell is thin (i.e., like a plane slab). The cylindrical shell of radii ri andro, length L and thermal conductivity k has the thermal resistance

Rt = ln(ro/ri )

2πkL(1.42)

Fins are extended surfaces that enhance the thermal contact between a base (Tb)and a fluid flow (T∞) that bathes the wall. If the geometry of the fin is such that thecross-sectional area (Ac), the wetted perimeter of the cross-section (p), and the heattransfer coefficient [h, defined later in Eq. (1.56)] are constant, the heat transferrate qb through the base of the fin is approximated well by

qb∼= (Tb − T∞)(k Achp)1/2 tanh

[(hp

k Ac

)1/2 (L + Ac

p

)](1.43)

In this expression L is the fin length measured from the tip of the fin to the basesurface, and k is the thermal conductivity of the fin material. The heat transfer ratethrough a fin with variable cross-sectional area and perimeter can be calculated bywriting

qb = (Tb − T∞)h Aexpη (1.44)

where Aexp is the total exposed (wetted) area of the fin and η is the fin efficiency, adimensionless number between 0 and 1, which can be found in heat transfer books[8, 9].

Equations (1.43) and (1.44) are based on the very important assumption that theconduction through the fin is essentially unidirectional and oriented along the fin.This assumption is valid when the Biot number Bi = ht/k is small such that [8](

ht

k

)1/2

< 1 (1.45)



22 Flow Systems

where t is the thickness of the fin, that is, the fin dimension perpendicular to theconduction heat current q.

The Biot number definition should not be confused with the Nusselt numberdefinition. The thermal conductivity k that appears in the Bi definition is theconductivity of the solid wall (e.g., fin) that is swept by the convective flow (h). Inthe Nu group defined later in Eq. (1.60), k is the thermal conductivity of the fluidin the convective flow.

Time-dependent conduction is also a phenomenon that we encounter and exploitfor design in this book. For example, the temperature distribution in a semi-infinitesolid, the surface temperature of which is raised instantly from Ti to T∞, is

T (x, t) − T∞Ti − T∞

= erf

[x

2(αt)1/2

](1.46)

The counterpart of this heating configuration is the surface with imposed heatflux. The temperature field under the surface of a semi-infinite solid that, startingwith the time t = 0, is exposed to a constant heat flux q′′ (heat transfer rate per unitarea) is

T (x, t) − Ti = 2q ′′

k

(αt

π

)1/2

exp

(− x2

4αt

)− q ′′

kx erfc

[x

2(αt)1/2

](1.47)

Note the arguments of erf, exp, and erfc: they all contain the group x/(αt)1/2, wherex is the distance (under the surface) to which the thermal wave penetrates duringthe time t, and α is the thermal diffusivity of the material (α = k/ρc). In Eq. (1.46),for example, thermal penetration means that x is such that T(x,t) – T∞ ∼ Ti – T∞,which means that

x

2(αt)1/2∼ 1 (1.48)

Therefore, a characteristic of all thermal diffusion processes is that the thicknessof thermal penetration under the exposed surface grows as (αt)1/2, that is infinitelyfast at t = 0+, and more slowly as t increases. We return to this characteristic in thetime-optimization of electrokinetic decontamination in Chapter 9.

Additional examples of time-dependent diffusion are the temperature fields thatdevelop around concentrated heat sources and sinks. Common phenomena thatcan be described in terms of concentrated heat sources are underground fissuresfilled with geothermal steam, underground explosions, canisters of nuclear andchemical waste, and buried electrical cables. It is important to distinguish betweeninstantaneous heat sources and continuous heat sources.

Consider first instantaneous heat sources released at t = 0 in a conductingmedium with constant properties (ρ, c, k, α) and uniform initial temperature Ti.The temperature distribution in the vicinity of the source depends on the shape of




the source:

T (x, t) − Ti = Q′′

2ρc(παt)1/2exp

(− x2

4αt

)[instantaneous plane source, strength Q′′(J/m2) at x = 0]

(1.49)

T (r, t) − Ti = Q′

4ρcπαtexp

(− r2

4αt

)[instantaneous line source, strength Q′(J/m) at r = 0] (1.50)

T (r, t) − Ti = Q

8ρc(παt)3/2exp

(− r2

4αt

)[instantaneous point source, strength Q(J) at r = 0] (1.51)

The temperature distributions near continuous heat sources, which release heatat constant rate when t > 0, are

T (x, t) − Ti = q ′′

ρc

(t

πα

)1/2

exp

(− x2

4αt

)− q ′′ |x |

2kerfc

[ |x |2 (αt)1/2

][continuous plane source, strength q ′′ (W/m2) at x = 0]

(1.52)

T (r, t) − Ti = q ′

4πk

∫ ∞

r2/4αt

e−u

udu ∼= q ′

4πk

[ln

(4αt

r2

)− 0.5772

], if

r2

4αt< 1

[continuous line source, strength q ′(W/m) at r = 0]

(1.53)

T (r, t) − Ti = q

4πkrerfc

[r

2(αt)1/2

]∼= q

4πkrif

r2

2 (αt)1/2 < 1

[continuous point source, strength q(W) at r = 0] (1.54)

Equations (1.49) through (1.54) also describe the temperature fields near concen-trated heat sinks. In such cases the numerical values of the source strengths (Q′′,Q′, Q, q ′′, q ′, q) are negative.

A related time-dependent configuration is plane melting and solidification. Asemi-infinite solid that is isothermal and at the melting point (Tm) melts if its surfaceis raised to a higher temperature (T0). In the absence of the effect of convection,the liquid layer thickness δ increases in time according to

δ(t) ∼=[

2kt

ρhs f(T0 − Tm)

]1/2

(1.55)

where ρ and k are the density and thermal conductivity of the liquid, and hsf is thelatent heat of melting. Equation (1.55) is valid provided that c(T0 – Tm)/hsf < 1,where c is the specific heat of the liquid.



24 Flow Systems

The solidification of a motionless pool of liquid is described by Eq. (1.55), inwhich T0 – Tm is replaced by Tm – T0 because the liquid is saturated at Tm, and thesurface temperature is lowered to T0. In the resulting expression δ is the thicknessof the solid layer, and k, c and α are properties of the solid layer.

1.3.2 Convection

Convection is the heat transfer mechanism in which a flowing material (gas, fluid,solid) acts as a conveyor for the energy that it draws from (or delivers to) a solid wall.As a consequence, the heat transfer rate is affected greatly by the characteristicsof the flow (e.g., velocity distribution, turbulence). To know the flow configurationand the regime (laminar vs. turbulent) is an important prerequisite for calculatingconvection heat transfer rates. Furthermore, the nature of the boundary layers(hydrodynamic and thermal) plays an important role in evaluating convection.

Convection is said to be external when a much larger space filled with flowingfluid (the free stream) exchanges heat with a body immersed in the fluid. Accordingto Eq. (1.38), the objective is to determine the relation between the heat transferrate (or the heat flux through a spot on the wall, q′′), and the wall-fluid temperaturedifference (Tw − T∞). The alternative is to determine the convective heat transfercoefficient h, which for external flow is defined by

h = q ′′

Tw − T∞(1.56)

where q′′ is the heat flux, q′′ = q/A, where A is the area swept by the flowing fluid.This means that Eq. (1.56) can also be written as

q = h A (Tw − T∞) (1.57)

or that the convective thermal resistance is

Rt = Tw − T∞q

= 1

h A(1.58)

Figure 1.12 shows the order of magnitude of h for various classes of convectiveheat transfer configurations. Techniques for estimating h accurately are available(e.g., Ref. [6]). The most basic example is the boundary layer on a plane wall.When the fluid velocity U∞ is uniform and parallel to a wall of length L, thehydrodynamic boundary layer along the wall is laminar over L if ReL ≤ 5 × 105,where the Reynolds number is defined by ReL = U∞L/ν and ν is the kinematicviscosity. The leading edge of the wall is perpendicular to the direction of the freestream (U∞). The wall shear stress in laminar flow averaged over the length L is

τ = 0.664ρU 2∞Re−1/2

L (ReL ≤ 5 × 105) (1.59)




Boiling, water

Boiling, organic liquids

Condensation, water vapor

Condensation, organic vapors

Liquid metals, forced convection

Water, forced convection

Organic liquids, forced convection

Gases, 200 atm, forced convection

Gases, 1 atm, forced convection

Gases, natural convection

1 10 102 103 104 105 106

h (W/m2 K)

Figure 1.12 Convective heat transfer coefficients, showing the effect of flow configuration [8].

so that the total tangential force experienced by a plate of width W and length Lis F = τ LW . The length L is measured in the flow direction. The thickness of thehydrodynamic boundary layer at the trailing edge of the plate is of order LRe−1/2

L .If the wall is isothermal at Tw, the heat transfer coefficient h averaged over the

flow length L is (ReL ≤ 5 × 105):

Nu = hL

k=

{0.664 Pr1/3 Re1/2

L (Pr ≥ 0.5)

1.128 Pr1/2 Re1/2L (Pr ≤ 0.5)

(1.60)

In these expressions k and Pr are the fluid thermal conductivity and the Prandtlnumber Pr = ν/α. The free stream is isothermal at T∞. The total heat transfer ratethrough the wall of area LW is q = hLW (Tw − T∞). The group Nu = hL/k is theoverall Nusselt number, where k is the thermal conductivity of the fluid.



26 Flow Systems

When the wall heat flux q′′ is uniform, the wall temperature Tw increases awayfrom the leading edge (x = 0) as x1/2:

Tw(x) − T∞ = 2.21q ′′xk Pr1/3 Re1/2

x

(Pr ≥ 0.5, Rex ≤ 5 × 105) (1.61)

Here, the Reynolds number is based on the distance from the leading edge, Rex

= U∞x/ν. The wall temperature averaged over the flow length L, Tw, is obtainedby substituting, respectively, Tw, ReL, and 1.47 in place of Tw(x), Rex, and 2.21 inEq. (1.61).

At Reynolds numbers ReL greater than approximately 5 × 105, the boundarylayer begins with a laminar section that is followed by a turbulent section. For 5 ×105 < ReL < 108 and 0.6 < Pr < 60, the average heat transfer coefficient h andwall shear stress τ are

Nu = hL

k= 0.037 Pr 1/3(Re4/5

L − 23,550) (1.62)

τ

ρU 2∞= 0.037

Re1/5L

− 871

ReL(1.63)

Equation (1.62) is sufficiently accurate for isothermal walls (Tw) as well as foruniform-heat flux walls. The total heat transfer rate from an isothermal wall isq = hLW (Tw − T∞), and the average temperature of the uniform-flux wall is Tw =T∞ + q ′′/h. The total tangential force experienced by the wall is F = τ LW , whereW is the wall width. The Nu results for many other external flow configurations(sphere, cylinder in cross-flow, etc.) are provided in Refs. [6, 9].

In flows through ducts, the heat transfer surface surrounds and guides the stream,and the convection process is said to be internal. For internal flows the heat transfercoefficient is defined as

h = q ′′

Tw − Tm(1.64)

Here, q′′ is the heat flux (heat transfer rate per unit area) through the wall wherethe temperature is Tw, and Tm is the mean (bulk) temperature of the stream,

Tm = 1

U A

∫ ∫A

uT dA (1.65)

The mean temperature is a weighted average of the local fluid temperature T overthe duct cross-section A. The role of weighting factor is played by the longitudinalfluid velocity u, which is zero at the wall and large in the center of the duct crosssection [e.g., Eq. (1.6)]. The mean velocity U is defined by

U = 1

A

∫ ∫A

u dA (1.66)




The volumetric flow rate through the A cross-section is

Q = U A =∫ ∫

Au dA (1.66′)

In laminar flow through a duct, the velocity distribution has two distinct regions:the entrance region, where the walls are lined by growing boundary layers, andfarther downstream, the fully developed region, where the longitudinal velocity isindependent of the position along the duct. It is assumed that the duct geometry(cross-section A, internal wetted perimeter p) does not change with the longitudinalposition. A measure of the length scale of the duct cross section is the hydraulicdiameter Dh defined in Eq. (1.21). The hydrodynamic entrance length X for laminarflow can be calculated with the formula (1.15), or, more exactly,

X

Dh∼ 0.05ReDh (1.67)

where the Reynolds number is based on mean velocity and hydraulic diameter,ReDh = UDh/ν. As shown in section 1.2.1, when the duct is much longer thanits entrance length, L � X, the laminar flow is fully developed along most of thelength L, and the friction factor is independent of L [cf. Eq. (1.23) and Table 1.2].

The heat transfer coefficient in fully developed flow is constant, that is, inde-pendent of longitudinal position. Table 1.2 lists the h values for fully developedlaminar flow for two heating models: duct with uniform heat flux (q′′) and ductwith isothermal wall (Tw). Using these h values is appropriate when the duct lengthL is considerably greater than the thermal entrance XT over which the temperaturedistribution is developing (i.e., changing) from one longitudinal position to thenext. The thermal entrance length for the entire range of Prandtl numbers is

XT

Dh∼ 0.05ReDh (1.68)

Note that in Table 1.2 the Nusselt number (Nu = hDh/k) is based on Dh, which isunlike in Eq. (1.60). Means for calculating the heat transfer coefficient for laminarduct flows in which XT is not much smaller than L can be found in Ref. [6].

Turbulent flow becomes fully developed hydrodynamically and thermally aftera relatively short entrance distance:

X ∼ XT ∼ 10Dh (1.69)

In fully developed turbulent flow (L � X) the friction factor f is independent of L,as shown by the family of curves drawn for turbulent flow on the Moody chart (Fig.1.3). The turbulent flow curves can be used for ducts with other cross-sectionalshapes, provided that D is replaced by the appropriate hydraulic diameter of theduct, Dh. The heat transfer coefficient h is constant in fully developed turbulentflow and can be estimated based on the Colburn analogy between heat transfer and



28 Flow Systems

Table 1.2 Friction factors (f ) and heat transfer coefficients (h) forhydrodynamically fully developed laminar flows through ducts [6].

h Dh/k

Cross-section shape Po = f ReDh Uniform q′′ Uniform Tw

13.3 3 2.35

14.2 3.63 2.89

16 4.364 3.66

18.3 5.35 4.65

24 8.235 7.54

24 5.385 4.86

momentum transfer:

h

ρcPU= f/2

Pr2/3 (1.70)

This formula holds for Pr ≥ 0.5 and is to be used in conjunction with Fig.1.3, which supplies the f value. Equation (1.70) applies to ducts of various cross-sectional shapes, with wall surfaces having uniform temperature or uniform heatflux and various degrees of roughness. The dimensionless group St = h/(ρcPU )is known as the Stanton number.

How does the temperature vary along a duct with heat transfer? Referring to Fig.1.13, the first law can be applied to an elemental duct length dx to obtain

dTm

dx= p

A

q ′′

ρcPU(1.71)

where p is the wetted internal perimeter of the duct and A is the cross-sectionalarea. Equation (1.71) holds for both laminar and turbulent flow. It can be combinedwith Eq. (1.64) and the h value furnished by Table 1.2 to determine the longitudinalvariation of the mean temperature of the stream, Tm(x). When the duct wall is




WallWall,

Stream, Tm(x) Stream, Tm(x)

T T

Tw

Tw

Tout

Tout

Tout

Tout

Tin

Tin

Tin

Tin

�

��

�

(x)

LL xx0

(a) (b)

0

Figure 1.13 Distribution of temperature along a duct with heat transfer: (a) isothermal wall;(b) wall with uniform heat flux [8].

isothermal at Tw (Fig. 1.13a), the total rate of heat transfer between the wall and astream with the mass flow rate m is

q = mcP�Tin

[1 − exp

(−h Aw

mcP

)](1.72)

where Aw is the wall surface, Aw = pL. A more general alternative to Eq. (1.72) is

q = h Aw�Tlm (1.73)

where �T lm is the log-mean temperature difference

�Tlm = �Tin − �Tout

ln (�Tin/�Tout)(1.74)

Equation (1.73) is more general than Eq. (1.72) because it applies when Tw isnot constant, for example, when Tw(x) is the temperature of a second stream incounterflow with the stream whose mass flow rate is m. Equation (1.72) can bededuced from Eqs. (1.73) and (1.74) by writing Tw = constant, and �T in = Tw –T in and �Tout = Tw − Tout. When the wall heat flux is uniform (Fig. 1.13b), thelocal temperature difference between the wall and the stream does not vary withthe longitudinal position x: Tw(x) − Tm(x) = �T (constant). In particular, �T in =�Tout = �T , and Eq. (1.74) yields �T lm = �T . Equation (1.73) reduces in thiscase to q = h Aw�T .

In natural convection, or free convection, the fluid flow is driven by the effectof buoyancy. This effect is distributed throughout the fluid and is associated withthe tendency of most fluids to expand when heated. The heated fluid becomes lessdense and flows upward, while packets of cooled fluid become more dense and sink.

One example is the natural convection boundary layer formed along an isother-mal vertical wall of temperature Tw, height H, and width W, which is in contactwith a fluid with far-field temperature T∞. Experiments show that the transition



30 Flow Systems

from the laminar section to the turbulent section of the boundary layer occurs at thealtitude y (between the leading edge, y = 0, and the trailing edge, y = H), where [8]

Ray ∼ 109 Pr (10−3 < Pr < 103) (1.75)

In this expression Ray is the Rayleigh number based on temperature difference,

Ray = gβ(Tw − T∞)y3

αν(1.76)

where β is the coefficient of volumetric thermal expansion, β = (−1/ρ) (∂ρ/∂T )P.If the fluid behaves as an ideal gas, β equals 1/T , where T is expressed in K. Theheat transfer results shown next are valid when |β(Tw − T∞)| � 1. The boundarylayer remains laminar over its entire height H when Ray < 109 Pr. The average heattransfer coefficient (h) for a wall of height H with laminar boundary layer flow is [6]:

h H

k=

{0.671Ra1/4

H , Pr � 1

0.8(RaH Pr)1/4, Pr � 1(1.77)

where RaH = gβ (Tw – T∞)H3/(αν). The heat transfer rate from the wall isq = h H W (T w − T∞), where W is the width (horizontal dimension) of the verticalwall.

A vertical wall that releases the uniform heat flux q′′ into the fluid has a temper-ature that increases with altitude, Tw(y). The relation between q′′ and the wall-fluidtemperature difference (Tw − T∞) is represented adequately by Eq. (1.77), providedh is replaced by q ′′/(Tw − T∞). Note that the temperature difference (Tw − T∞) isaveraged over the wall height H.

We end with examples of chimney flow: fully developed laminar flow drivenby buoyancy through a vertical duct of hydraulic diameter Dh, height H, and innersurface temperature Tw. The top and bottom ends of the duct are open to a fluid ata temperature T∞. It is assumed that the duct is sufficiently slender so that H/Dh >

RaDh , where RaDh = gβ(Tw − T∞)D3h/αν. The average heat transfer coefficient h

depends on the shape of the duct cross section:

h H/k

RaDh

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1

192(parallel plates)

1

128(round)

1

113.6(square)

1

106.4(equilaterial triangle)

(1.78)

The total heat transfer rate between the duct and the stream is approximately q ∼=mcP (Tw − T∞) when the group h Aw/mcp is greater than 1, where the mass flowrate is m = ρ AU , and the duct cross-sectional area is A. The mean velocity U can



Problems 31

be estimated using Eq. (1.22), in which �P/L is now replaced by ρgβ(Tw − T∞).In other words, the chimney flow is Poiseuille flow driven upward by the effectivevertical pressure gradient ρgβ(Tw − T∞).

More examples of thermal resistance dominate the field of radiation heat transfer.For this body of thermal sciences we refer the reader to more complete treatments(e.g. Refs. [8, 9]).

Mass transfer is the transport of chemical species in the direction from highspecies concentrations to lower concentrations. Mass diffusion is analogous tothermal diffusion (section 1.3.1); therefore, it is not expanded in this section. In thesimplest treatment, instead of the Fourier law (1.39), mass diffusion is based onthe Fick law (Chapter 9), which proclaims a proportionality between species massflux and concentration gradient. This topic is treated in detail in Chapter 9.

REFERENCES

1. A. Bejan, Advanced Engineering Thermodynamics, 2nd ed. (Ch. 13). New York: Wiley,1997.

2. A. Bejan, Shape and Structure, from Engineering to Nature. Cambridge, UK: CambridgeUniversity Press, 2000.

3. A. Bejan and S. Lorente, The constructal laws and the thermodynamics of flow systemswith configuration. Int J Heat Mass Transfer, Vol. 47, 2004, pp. 3073–3083.

4. A. Bejan and S. Lorente, La Loi Constructale. Paris: L’Harmattan, 2005.

5. S. Lorente and A. Bejan, Svelteness, freedom to morph, and constructal multi-scale flowstructures. Int J Thermal Sciences, Vol. 44, 2005, pp. 1123–1130.

6. A. Bejan, Convection Heat Transfer, 3rd ed. Hoboken, NJ: Wiley, 2004.

7. A. Bejan, G. Tsatsaronis, and M. Moran, Thermal Design and Optimization. New York:Wiley, 1996.

8. A. Bejan, Heat Transfer. New York: Wiley, 1993.

9. A. Bejan and A. D. Kraus, eds., Heat Transfer Handbook. Hoboken, NJ: Wiley, 2003.

10. A. Bejan, Advanced Engineering Thermodynamics, 3rd ed. Hoboken: Wiley, 2006.

11. A. Bejan and M. Almogbel, Constructal T-shaped fins. Int J Heat Mass Transfer, Vol. 43,2000, pp. 141–164.

12. G. Lorenzini and S. Moretti, Numerical analysis of heat removal enhancement with extendedsurfaces. Int J Heat Mass Transfer, Vol. 50, 2007, pp. 746–755.

13. A. Bejan, How to distribute a finite amount of insulation on a wall with nonuniformtemperature. Int J Heat Mass Transfer, Vol. 36, 1993, pp. 49–56.

PROBLEMS

1.1. Flow strangulation is not good for performance. Uniform distribution ofstrangulation is. Consider a long duct with Poiseuille flow (Fig. P1.1). Theduct has two sections, a narrow one of length L1 and cross-sectional area A1,followed by a wider one of length L2 and cross-sectional area A2. The total



32 Flow Systems

length L, the mass flow rate, and the total duct volume are fixed. Show thatthe duct with minimal global flow resistance is the one with uniform cross-section (A1 = A2). To demonstrate this analytically, write that the pressuredrop along each duct section is proportional to the flow length and inverselyproportional to the cross-sectional area squared.

A1A2 A2

m m

L1

L

�P

� �

Figure P1.1

1.2. Consider the following heat conduction illustration (Fig. P1.2). A long barof heat-conducting material (thermal conductivity k, length L) has two sec-tions: one thin (length L1, cross-section A1) and the other thicker (length L2,cross-section A2). The total volume of conducting material (V) is fixed.Heat is conducted in one direction (along the bar) in accordance withthe Fourier law. The global thermal resistance of the bar is �T/q, where�T is the temperature drop along the distance L, and q is the end-to-endheat current. Show analytically that �T/q is minimum when A1 = A2, thatis, optimal distribution of imperfection means uniform distribution of flowstrangulation.

A2 A2A1

L1

L

�T

q q

Figure P1.2

1.3. The phenomenon of self-lubrication is an illustration of the natural tendencyof flow systems to generate configurations that provide maximum access forthe things that flow. Consider the two-dimensional configuration shown inFig. P1.3. The sliding solid body has an unspecified thickness D. The slotthrough which the body slides has the spacing D + 2δ. The relative-motiongaps have the spacings δ + ε and δ – ε, where ε represents the eccentricityof the body-housing alignment. The relative-motion gaps are filled with a



Problems 33

Newtonian liquid of viscosity μ. The body slides parallel to itself with thevelocity V . Show that the body encounters minimum resistance when itcenters itself in the slot (ε = 0).

μ

μ

δ−ε

δ+ε

V

Figure P1.3

1.4. A round rod slides through a round housing as shown in Fig. P1.4. Themotion of the rod is in the direction perpendicular to the figure. In general,the rod and housing cross-sections are not concentric. The eccentricity ε andthe gap thickness δ(θ ) are much smaller than the rod radius R. The gap isfilled with a lubricating liquid of viscosity μ. The rod slides with the velocityV . Show that the configuration that obstructs the movement of the rod theleast is the concentric configuration (ε = 0). When cylindrical bodies slidewith lubrication and coat themselves with a liquid film of constant thickness,they exhibit the natural phenomenon of self-lubrication. The concentricconfiguration, which is predictable from the maximization of flow access, isa constructal configuration.

Figure P1.4

1.5. Two Newtonian liquids flow coaxially in the Poiseuille regime through a tubeof radius R. The liquids have the same density ρ. The longitudinal pressuregradient (–dP/dx) is specified. The liquids have different viscosities, μ1 >

μ2. One liquid flows through a central core of radius Ri, and the other flows



34 Flow Systems

through an annular cross-section extending from r = Ri to the wall (r = R).Show that the total flow rate is greater when the μ2 liquid coats the wall.This phenomenon of self-lubrication is observed in nature and is demandedby the constructal law. Self-lubrication is achieved through the generation offlow configuration in accordance with the constructal law.

1.6. The modern impulse turbine developed based on the Pelton wheel inventionhas buckets in which water jets are forced to make 180◦ turns. According tothe simple model shown in Fig. P1.6, nozzles bring to the wheel a streamof water of mass flow rate m, room temperature, room pressure, and inletvelocity V in (fixed). The bucket peripheral speed is V . The relative watervelocity parallel to V , in the frame attached to the bucket, is Vr. The outletstream velocity in the stationary frame is Vout. Determine the optimal wheelspeed V such that the shaft power generated by the wheel is maximum.

A, viewed in the radial direction

A

W

V

Vr

m , Vout

m , Vin

m/2

m/2

m

.

.

.

.

.

.

Figure P1.6

1.7. In this problem we explore the idea to replace the two-dimensional designof an airplane wing or wind mill blade with a three-dimensional design thathas bulbous features on the front of the wing (Fig. P1.7) [10]. If the frontalportion of the wing is approximated by a long cylinder of diameter Dc, theproposal is qualitatively the same as replacing this material with severalequidistant spheres of diameter Ds and spacing S. The approach air velocityis V . In the Reynolds number range 103 < VDc,s/ν < 105, the sphere andcylinder drag coefficients are Cs

∼= 0.4 and Cc∼= 1, (cf. Fig. 1.11). Evaluate

the merit of this proposal by comparing the drag forces experienced by thespheres and cylinder segment, alone (i.e., by neglecting the rest of the wing).



Problems 35

Bulbous front

Two-dimensionalfront

Dc

S

V

S

Dc

Ds

Figure P1.7

1.8. Show that the scale of the residence time of water in a river basin is propor-tional to the scale of the flow path length divided by the gradient (slope) ofthe flow path [10]. A simple and effective way to demonstrate this is to relyon constructal theory, according to which “optimal distribution of imperfec-tion” in river basins means a balance between the high-resistance seepagealong the hill slope and the low-resistance channel flow. Consequently, theglobal scales of the river basin are comparable with the scales of Darcy-flowseepage along the hill slope of length L, slope z/L, and water travel time t.Show that the proportionality between t and L/(z/L) is t ∼ (ν /gK)L/(z/L),where ν is the water kinematic viscosity and g is the gravitational accelera-tion. The permeability of soil K has values in the range 2.9 × 10−9 to 1.4 ×10−7 cm2.

1.9. The time required to inflate a tire or a basketball is shorter than the timeneeded by the compressed air to leak out if the valve is left open. Model theinhaling-exhaling cycle executed by the lungs, and explain why the exhalingtime is longer than the inhaling time, even though the two times are onthe same order of magnitude. A simple one-dimensional model is shownin Fig. P1.9. The lung volume expands and contracts as its contact surfacewith the thorax travels the distance L. The thorax muscles pull this surfacewith the force F. The pressure inside the lung is P, and outside is Patm.The flow of air into and out of the lung is impeded by the flow resistancer, such that Patm − P = rmn

in during inhaling and P − Patm = rmnout during

exhaling. The exponent n is a number between n ∼= 2 (turbulent flow) and



36 Flow Systems

n = 1 (laminar flow). For simplicity, assume that the density of air is constantand that the volume expansion rate during inhaling (dx/dt) is constant. Thelung tissue is elastic and can be modeled as a linear spring that places therestraining force kx on surface A. Determine analytically the inhaling time(t1) and the exhaling time (t2), and show that t2/t1 ≥ 2.

Exhaling

Lungs Lungs Patm

r r P

L x

0 L x

0

P

A A

F

k k

Patm Patm Patm

Inhaling

Thorax min . mout

.

Figure P1.9

1.10. A domestic refrigerator and freezer have two inner compartments, the re-frigerator or cooler of temperature Tr and the freezer of temperature Tf . Theambient temperature is TH . This assembly is insulated with a fixed amountof insulation (Af Lf + ArLr = constant), where Af and Ar are the surfaces ofthe freezer and the refrigerator enclosures. The thermal conductivity of theinsulating material is k. Figure P1.10 shows this assembly schematically, inunidirectional heat flow fashion, with cold on the left, and warm on the right.

Freezer

Refrigerator(Cooler)

Af , Tf

Ar , Tr

TH

TH

Lf

Lr

k

k

Figure P1.10



Problems 37

The two heat currents that penetrate the insulation and arrive at Tf and Tr areremoved by a reversible refrigerating machine that consumes the power W.Determine the optimal insulation architecture, for example, the ratio Lf /Lr,such that the total power requirement W is minimum.

1.11. As shown in Fig. P1.11, a hot stream originally at the temperature Th flowsthrough an insulated pipe suspended in a space at temperature T0. The streamtemperature T(x) decreases in the longitudinal direction because of the heattransfer that takes place from T(x) to T0 everywhere along the pipe. Thefunction of the pipe is to deliver the stream at a temperature (Tout) as closeas possible to the original temperature Th. The amount of insulation is fixed.Determine the best way of distributing the insulation along the pipe.

Tout

T0, Ambient

Pipe wall

Hot stream

Insulation

0 x

hT,m

k

k

t(x)

Figure P1.11

1.12. The temperature distribution along the two-dimensional fin with sharp tipshown in Fig. P1.12 is linear, θ (x) = (x/L) θb, where the local temperaturedifference is θ (x) = T (x) − T∞ and θb = Tb − T∞. The fin temperature isT(x), base temperature is Tb, and the fluid temperature is T∞. The fin widthis W. The tip is at the temperature of the surrounding fluid: θ (0) = 0. Showthat the thickness δ of this fin must be δ = (h/k) x2. Derive an expressionfor the total heat transfer rate through the base of this fin, Qb.

Figure P1.12

1.13. The sizing procedure for a fin generally requires the selection of morethan one physical dimension. In a plate fin, Fig. P1.13, there are two such



38 Flow Systems

dimension, the length of the fin, L, and the thickness, t. One of these dimen-sions or the shape of the fin profile t/L can be selected optimally when thetotal volume of fin material is fixed. Such a constraint is often justified bythe high cost of the high-thermal-conductivity metals that are employed inthe manufacture of finned surfaces (e.g., copper, aluminum) and by the costassociated with the weight of the fin.

Consider a plate fin of constant thickness t, conductivity k, and length Lextending from a wall of temperature Tb into a fluid flow of temperature T∞[8]. The plate-fin dimension perpendicular to t × L is W. The heat transfercoefficient h is uniform on all the exposed surfaces. The profile is slender(t � L), the Biot number is small (ht/k � 1), and the unidirectional finconduction model applies. Assume that the heat transfer through the fintip is negligible so that the total heat transfer rate through the fin base isqb = θb (k Achp)1/2 tanh(mL), where the notation is θb = (Tb − T∞), m =(hp/k Ac)

1/2, Ac = tW , and p = 2W. The global conductance of the fin isqb/θb. Maximize this function subject to the constraint that the fin volume isfixed, V = tLW. Report analytically the optimal shape of the fin profile andthe maximized global conductance.

ht k

L

t

0.077 0.032 0.01 �

Figure P1.13

1.14. Consider the T-shaped assembly of fins [11, 12] sketched in Fig. P1.14.Two elemental fins of thickness t0 and length L0 serve as tributaries to astem of thickness t1 and length L1. The configuration is two-dimensional,with the third dimension (W) sufficiently long in comparison with L0 andL1. The heat transfer coefficient h is uniform over all the exposed surfaces.The temperatures of the root (T1) and the fluid (T∞) are specified. Thetemperature at the junction (T0) is one of the unknowns and varies with thegeometry of the assembly.

The objective is to determine the optimal geometry (L1/L0, t1/t0) that ischaracterized by the maximum global thermal conductance q1/(T1 – T∞),where q1 is the heat current through the root section. The optimization issubjected to two constraints, the total-volume (i.e., front area) constraint,



Problems 39

A = 2L0L1, and the fin-material volume constraint, Af = 2L0t0 + t1L1. Thelatter can be expressed in dimensionless form as the solid volume fractionφ1 = Af /A, which is a specified constant considerably smaller than 1. Theanalysis that delivers the global conductance as a function of the assemblygeometry consists of accounting for conduction along the L0 ad L1 fins andinvoking the continuity of temperature and heat current at the T junction.The unidirectional conduction model is recommended for the analysis ofeach fin. Develop the global conductance as the dimensionless functionq1(L0,t0,L1,t1,a), where

q1 = q1

kpW (T1 − T∞)a =

(h A1/2

kp

)1/2

kp is the fin material thermal conductivity and (L0,t0,L1,t1) =(L0,t0,L1,t1) /A1/2

1 . Set a = 1 and φ1 = 0.1. Use the dimensionless ver-sions of the A and Af constraints and recognize that the geometry of theassembly has only two degrees of freedom. Choose L1/L0 and t1/t0 as de-grees of freedom. First, maximize numerically q1 with respect to L1/L0 whileholding t1/t0 constant, and your result will be the curve q1,m(t1/t0). Finally,maximize q1,m with respect to t1/t0 and report the twice-maximized con-ductance q1,mm and the corresponding architecture (L1/ L0)opt and (t1/t0) opt.Make a scale drawing of the T-shaped geometry that represents this design.

H1

W

L0

L1

T1

T0

q0

q1

t0

t1

q0

h, T�

h, T�

Figure P1.14

1.15. In this problem we consider the fundamental question of spreading opti-mally a finite amount of insulation on a nonisothermal surface. The simplest



40 Flow Systems

configuration in which this question can be examined is shown in Fig. P1.15.A plane wall of length L and width W (perpendicular to the figure) hasthe known temperature distribution T(x). A layer of insulating material oflow thermal conductivity k and unknown thickness t(x) covers the wall. Thetemperature of the ambient is T0. For simplicity, we assume that the thermalresistance (e.g., convection) between the outer surface of the insulation andthe ambient is negligible relative to the resistance of the insulating layer.This is equivalent to assuming that the temperature of the outer surface ofthe insulating layer is essentially uniform and equal to T0. When the layeris relatively thin (t � L), the flow of heat through the insulating material isunidirectional, perpendicular to the wall. Develop an integral expression forthe total heat leak through the arbitrary-thickness design t(x), and minimizedit subject to the constraint that the total volume of insulation is fixed. Relyon variational calculus (see Appendix C) to show that the optimal insulationthickness function is

topt(x) = tavgL[T (x) − T0]1/2∫ L0 [T (x) − T0]1/2dx

where tavg is the L-averaged thickness of the insulation. Determine the corre-sponding (minimal) heat leak through the insulation, qmin. The same topt(x)distribution is obtained when the amount of insulation material is minimizedsubject to a fixed rate of heat loss to the ambient. Show that when T(x) in-creases linearly from T0 at x = 0 to T0 + �T at x = L, the optimal thicknessincreases as x1/2 along the wall. This problem was proposed and treated indetail in Ref. [13].

x = 0 x = LT(x), wall surface

Insulation, kAmbient, T0

t(x)

Figure P1.15

1.16. Consider again the problem of distributing a finite amount of insulationon a nonisothermal wall. In the preceding problem, we derived the generalsolution for the optimal distribution of insulation thickness and the corre-sponding total heat leak through the insulation, qmin. The example was fora wall with linear temperature distribution. In this problem, assume that thewall temperature T(x) has the general form

T (x) = T0 + �T( x

L

)n



Problems 41

where the end excess temperature �T is fixed, and n is a positive constant.Determine qmin as a function of n. In addition, consider the case whereinsulation thickness is uniform (t = tavg), and determine the correspondingheat leak (q), as a function of n. Show that the ratio qmin(n)/q(n) is equal to(n + 1)/( n

2 + 1)2, and that it decreases as n increases. Comment on the typeof wall temperature distributions for which the use of the optimal thicknessdesign is attractive.



42

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